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Sectional curvature of polygonal complexes with planar substructures.

Keller, Matthias and Peyerimhoff, Norbert and Pogorzelski, Felix (2017) 'Sectional curvature of polygonal complexes with planar substructures.', Advances in mathematics., 307 . pp. 1070-1107.


In this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus on the case of non-positive and negative combinatorial curvature. As geometric results we obtain a Hadamard–Cartan type theorem, thinness of bigons, Gromov hyperbolicity and estimates for the Cheeger constant. We employ the latter to get spectral estimates, show discreteness of the spectrum in the sense of a Donnelly–Li type theorem and present corresponding eigenvalue asymptotics. Moreover, we prove a unique continuation theorem for eigenfunctions and the solvability of the Dirichlet problem at infinity.

Item Type:Article
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Publisher statement:© 2016 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (
Date accepted:24 October 2016
Date deposited:25 October 2016
Date of first online publication:06 December 2016
Date first made open access:No date available

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